The principal inverse of the gamma function

Transcription

1 The principal inverse of the gamma function Mitsuru Uchiyama Shimane University 2013/7 Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 1 / 25

2 Introduction The gamma function Γ(x) is usually defined by the Euler form Γ(x) = 0 e t t x 1 dt for x > 0. This is extended to Rz > 0. By Γ(z + 1) = zγ(z) for z 0, 1, 2, it is defined and holomorphic on C \ {0, 1, 2, } Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 2 / 25

3 Introduction The gamma function Γ(x) is usually defined by the Euler form Γ(x) = 0 e t t x 1 dt for x > 0. This is extended to Rz > 0. By Γ(z + 1) = zγ(z) for z 0, 1, 2, it is defined and holomorphic on C \ {0, 1, 2, } The Weierstrass form 1 Γ(x) = xeγx (1 + x n )e x n (1) n=1 is useful, where γ is the Euler constant defined by γ = lim (1 + 1 n log n) = n Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 2 / 25

4 Introduction From (1) (Weierstrass form) it follows that log Γ(x) = log x γx + Γ (x) Γ(x) on C \ {0, 1, 2, }. = γ + n=1 ( x n log(1 + x n ) ), 1 ( n ) (psifunction) (2) n + x n=0 Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 3 / 25

5 Introduction From (1) (Weierstrass form) it follows that log Γ(x) = log x γx + Γ (x) Γ(x) = γ + n=1 ( x n log(1 + x n ) ), 1 ( n ) (psifunction) (2) n + x n=0 on C \ {0, 1, 2, }. By (2), Γ (z) Γ(z) maps the open upper half plane Π + into itself, namely Γ (z) Γ(z) is a Pick (Nevanlinna) function. Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 3 / 25

6 Introduction From (1) (Weierstrass form) it follows that log Γ(x) = log x γx + Γ (x) Γ(x) = γ + n=1 ( x n log(1 + x n ) ), 1 ( n ) (psifunction) (2) n + x n=0 on C \ {0, 1, 2, }. By (2), Γ (z) Γ(z) maps the open upper half plane Π + into itself, namely Γ (z) Γ(z) is a Pick (Nevanlinna) function. Γ (z) does not vanish on Π +. Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 3 / 25

7 Introduction Figure : Gamma function Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 4 / 25

8 Introduction Γ(1) = Γ(2) = 1, Γ (1) = γ, Γ (2) = γ + 1. Denote the unique zero in (0, ) of Γ (x) by α. α = , Γ(α) = Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 5 / 25

9 Introduction Γ(1) = Γ(2) = 1, Γ (1) = γ, Γ (2) = γ + 1. Denote the unique zero in (0, ) of Γ (x) by α. α = , Γ(α) = We call the inverse function of the restriction of Γ(x) to (α, ) the principal inverse function and write Γ 1. Γ 1 (x) is an increasing and concave function defined on (Γ(α), ). Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 5 / 25

10 Introduction Main Theorem Theorem 1 The principal inverse Γ 1 (x) of Γ(x) has the holomorphic extension Γ 1 (z) to D := C \ (, Γ(α)], which satisfies (i) Γ 1 (Π + ) Π + and Γ 1 (Π ) Π, (ii) Γ 1 (z) is univalent, (iii) Γ(Γ 1 (z)) = z for z D. Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 6 / 25

11 kernel function Definition 2 Let I be an interval in R and K(x, y) a continuous function defined on I I. Then K(x, y) is said to be a positive semidefinite (p.s.d.) kernel function on an interval I I (on I for short) if K(x, y)ϕ(x)ϕ(y)dxdy 0 (3) I I for every complex continuous function ϕ with compact support in I. Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 7 / 25

12 kernel function Definition 2 Let I be an interval in R and K(x, y) a continuous function defined on I I. Then K(x, y) is said to be a positive semidefinite (p.s.d.) kernel function on an interval I I (on I for short) if K(x, y)ϕ(x)ϕ(y)dxdy 0 (3) I I for every complex continuous function ϕ with compact support in I. K(x, y) is p.s.d. if and only if for each n and for all n points x i I, for n complex numbers z i. n K(x i, x j )z i z j 0 i,j=1 Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 7 / 25

13 kernel function Definition 3 K(x, y) is said to be conditionally (or almost) positive semidefinite (c.p.s.d.) on I I (on I for short) if (3) holds for every continuous function ϕ on I such that the support of ϕ is compact and I ϕ(x)dx = 0. K(x, y) is said to be conditionally negative semidefinite (c.n.s.d.) on I if K(x, y) is c.p.s.d. Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 8 / 25

14 kernel function Definition 3 K(x, y) is said to be conditionally (or almost) positive semidefinite (c.p.s.d.) on I I (on I for short) if (3) holds for every continuous function ϕ on I such that the support of ϕ is compact and I ϕ(x)dx = 0. K(x, y) is said to be conditionally negative semidefinite (c.n.s.d.) on I if K(x, y) is c.p.s.d. K(x, y) is c.p.s.d. if and only if n K(x i, x j )z i z j 0 (4) i,j=1 for each n, for all n points x i I and for n complex numbers z i with n i=1 z i = 0. Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 8 / 25

15 kernel function Facts K(x, y) = f (x)f (y) is p.s.d. If K(x, y) is p.s.d. on [a, b] and h : [c, d] [a, b] is increasing and (differentiable), then so is K(h(t), h(s)) on [c, d]. If K t (x, y) is p.s.d. for each t, then so is K t (x, y)dµ(t). (Schur) If K 1 (x, y) and K 2 (x, y) are both p.s.d. on I, then so is the product K 1 (x, y)k 2 (x, y). If K(x, y) is p.s.d. on I, then K(x, y) is c.p.s.d. on I. K(x, y) = x + y is not p.s.d. but c. p. s. d. and c. n. s. d. on any I. Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 9 / 25

16 kernel function Definition 4 Suppose K(x, y) 0 for every x, y in I. Then K(x, y) is said to be infinitely divisible on I I (on I for short) if K(x, y) a is positive semi-definite for every a > 0. Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 10 / 25

17 kernel function Definition 4 Suppose K(x, y) 0 for every x, y in I. Then K(x, y) is said to be infinitely divisible on I I (on I for short) if K(x, y) a is positive semi-definite for every a > 0. K(x, y) is infinitely divisible if and only if for each n, for all n points x i I and for every a > 0 matrix (K(x i, x j ) a ) is positive semi-definite. Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 10 / 25

18 kernel function Cauchy kernel 1 x + y is infinitely divisible on (0, ) (0, ). Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 11 / 25

19 kernel function Cauchy kernel 1 x + y is infinitely divisible on (0, ) (0, ). 1 (x + y) a = 1 Γ(a) 0 e tx e ty t a 1 dt Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 11 / 25

20 kernel function Lemma 5 (Fitzgerald, Horn) Let K(x, y) > 0 for x, y I and suppose K(x, y) is 1 c.p.s.d. on I I. Then exp( K(x, y)) and the reciprocal function K(x,y) are infinitely divisible there. Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 12 / 25

21 kernel function Lemma 5 (Fitzgerald, Horn) Let K(x, y) > 0 for x, y I and suppose K(x, y) is 1 c.p.s.d. on I I. Then exp( K(x, y)) and the reciprocal function K(x,y) are infinitely divisible there. Example K(x, y) := x + y on (0, ) (0, ) K(x, y) > 0 and K(x, y) is c.p.s.d. on (0, ) (0, ). (exp( K(x, y))) a = e ax e ay is p.s.d. for every a > 0. 1 K(x,y) = 1 x+y is the Cauchy kernel. Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 12 / 25

22 The Löwner kernel kernel function Definition 6 Let f (x) be a real C 1 -function on I. Then the Löwner kernel function is defined by { f (x) f (y) K f (x, y) = x y (x y) f (x) (x = y). Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 13 / 25

23 The Löwner kernel kernel function Definition 6 Let f (x) be a real C 1 -function on I. Then the Löwner kernel function is defined by { f (x) f (y) K f (x, y) = x y (x y) f (x) (x = y). Example (i) For f (x) = x, K f (x, y) = 1 is p. s. d. on R 2. (ii) For f (x) = 1 x+λ, K 1 f (x, y) = (x+λ)(y+λ) is p. s. d. on ( λ, ) ( λ, ) and on (, λ) (, λ). (iii) For f (x) = x 2, K f (x, y) = x + y is not p. s. d. but c. p. s. d. and c. n. s. d. Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 13 / 25

24 kernel function (Löwner Theorem) (also Koranyi) Let f (x) be a real C 1 -function on I. Then the Löwner kernel function K f (x, y) is p.s.d. on I I if and only if f (x) has a holomorphic extension f (z) to Π + which is a Pick (Nevanlinna) function. Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 14 / 25

25 kernel function (Löwner Theorem) (also Koranyi) Let f (x) be a real C 1 -function on I. Then the Löwner kernel function K f (x, y) is p.s.d. on I I if and only if f (x) has a holomorphic extension f (z) to Π + which is a Pick (Nevanlinna) function. We will show K Γ 1(x, y) is p. s. d. on (Γ(α), ) (Γ(α), ). Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 14 / 25

26 Proof Lemma 7 (Known result) K log x (x, y) := is p.s.d. on (0, ) (0, ) { log x log y x y (x y) 1 x (x = y) Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 15 / 25

27 Proof Lemma 7 (Known result) K log x (x, y) := is p.s.d. on (0, ) (0, ) Proof. By the formula { log x log y x y (x y) 1 x (x = y) log x = 0 ( 1 x+t + t )dt (x > 0), t 2 +1 Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 15 / 25

28 Proof Lemma 7 (Known result) K log x (x, y) := is p.s.d. on (0, ) (0, ) Proof. By the formula { log x log y x y (x y) 1 x (x = y) we obtain for x, y > 0. log x = 0 ( 1 x+t + t )dt (x > 0), t 2 +1 K log x (x, y) = 0 1 (x+t)(y+t) dt Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 15 / 25

29 Proof Lemma 8 K log Γ(x) (x, y) := K log Γ(x) (x, y) is c.p.s.d. on (0, ). { log Γ(x) log Γ(y) x y (x y) Γ (x) Γ(x) (x = y). Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 16 / 25

30 Proof Lemma 8 K log Γ(x) (x, y) := K log Γ(x) (x, y) is c.p.s.d. on (0, ). { log Γ(x) log Γ(y) x y (x y) Γ (x) Γ(x) (x = y). Proof. log Γ(x) = log x γx + ( x n=1 n log(1 + x n )) Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 16 / 25

31 Proof Lemma 8 K log Γ(x) (x, y) := K log Γ(x) (x, y) is c.p.s.d. on (0, ). { log Γ(x) log Γ(y) x y (x y) Γ (x) Γ(x) (x = y). Proof. log Γ(x) = log x γx + ( x n=1 n log(1 + x n )) g(x) := ( x k=1 k log(1 + x k )) log Γ(x) = log x γx + g(x) K log Γ(x) (x, y) = K log x (x, y) + γ K g (x, y). Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 16 / 25

32 Proof K log Γ(x) (x, y) = K log x (x, y) + γ K g (x, y). K log x (x, y) is p.s.d. and a constant function γ is c.p.s.d. We will see K g (x, y) is c.p.s.d. Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 17 / 25

33 Proof K log Γ(x) (x, y) = K log x (x, y) + γ K g (x, y). K log x (x, y) is p.s.d. and a constant function γ is c.p.s.d. We will see K g (x, y) is c.p.s.d. g n (x) := n k=1 ( x k log(1 + x k )). Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 17 / 25

34 Proof K log Γ(x) (x, y) = K log x (x, y) + γ K g (x, y). K log x (x, y) is p.s.d. and a constant function γ is c.p.s.d. We will see K g (x, y) is c.p.s.d. g n (x) := n k=1 ( x k log(1 + x k )). (i) K gn (x, y) is c.p.s.d. Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 17 / 25

35 Proof K log Γ(x) (x, y) = K log x (x, y) + γ K g (x, y). K log x (x, y) is p.s.d. and a constant function γ is c.p.s.d. We will see K g (x, y) is c.p.s.d. g n (x) := n ( x k=1 k log(1 + x k )). (i) K gn (x, y) is c.p.s.d. (ii) g n(x) = n k=1 [0, M]. x k(k+x) k=1 x k(k+x) = g (x) on any finite interval Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 17 / 25

36 Proof K log Γ(x) (x, y) = K log x (x, y) + γ K g (x, y). K log x (x, y) is p.s.d. and a constant function γ is c.p.s.d. We will see K g (x, y) is c.p.s.d. g n (x) := n ( x k=1 k log(1 + x k )). (i) K gn (x, y) is c.p.s.d. (ii) g n(x) = n k=1 [0, M]. x k(k+x) k=1 (iii) K gn (x, y) K g (x, y) = x k(k+x) = g (x) on any finite interval 1 x x y y (g n(t) g (t)) dt (x y) g n(x) g (x) (x = y) Therefore, K g (x, y) is c.p.s.d. on (0, ). Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 17 / 25

37 Proof Lemma 9 is infinitely divisible on (α, ). 1 K log Γ(x) (x, y) Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 18 / 25

38 Proof Lemma 9 is infinitely divisible on (α, ). Proof By the previous lemma 1 K log Γ(x) (x, y) K log Γ(x) (x, y) > 0 is c.n.s.d. on (α, ). By Fitzgerald and Horn s result, we get the required result. Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 18 / 25

39 Proof Lemma 10 Let K 1 (x, y) be the kernel function defined on (α, ) (α, ) by K 1 (x, y) = Then K 1 (x, y) is p.s.d. on (α, ). { x y Γ(x) Γ(y) (x y) (x = y). 1 Γ (x) Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 19 / 25

40 Proof Lemma 10 Let K 1 (x, y) be the kernel function defined on (α, ) (α, ) by K 1 (x, y) = Then K 1 (x, y) is p.s.d. on (α, ). Proof. K 1 (x, y) = { x y { log Γ(x) log Γ(y) Γ(x) Γ(y) 1 Γ(x) = K log x (Γ(x), Γ(y)) Γ(x) Γ(y) (x y) (x = y). 1 Γ (x) x y log Γ(x) log Γ(y) (x y) Γ(x) Γ (x) (x = y) 1 K log Γ(x) (x, y) Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 19 / 25

41 Proof of Theorem Proof We have shown K 1 (x, y) = is p.s.d. on (α, ). Hence { x y Γ(x) Γ(y) (x y) (x = y). 1 Γ (x) K Γ 1(x, y) = { Γ 1 (x) Γ 1 (y) x y (x y) (Γ 1 ) (x) (x = y) = K 1(Γ 1 (x), Γ 1 (y)) is p.s.d. on (Γ(α), ) (Γ(α), ). Thus by the Löwner theorem, Γ 1 (x) has the holomorphic extension Γ 1 (z) onto Π +, which is a Pick function. By reflection Γ 1 (x) has also holomorphic extension to Π and the range is in it. Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 20 / 25

42 Proof Γ(Γ 1 (z)) is thus holomorphic on the simply connected domain D := C \ (, Γ(α)], and Γ(Γ 1 (x)) = x for Γ(α) < x <. By the unicity theorem, Γ ( Γ 1 (z) ) = z for z D. It is clear that Γ 1 (z) is univalent. Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 21 / 25

43 Proof Γ(Γ 1 (z)) is thus holomorphic on the simply connected domain D := C \ (, Γ(α)], and Γ(Γ 1 (x)) = x for Γ(α) < x <. By the unicity theorem, Γ ( Γ 1 (z) ) = z for z D. It is clear that Γ 1 (z) is univalent. Corollary 11 There is a Borel measure µ so that where Γ(α) Γ 1 (x) = a + bx + Γ(α) 1 ( t x t t 2 )dµ(t), (5) dµ(t) <, and a, b are real numbers and b 0. t 2 +1 Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 21 / 25

44 Matrix inequality Application Theorem 12 The principal inverse Γ 1 (x) of Γ(x) is operator monotone on [Γ(α), ); i.e., and hence for bounded self-adjoint operators A, B whose spectra are in [Γ(α), ) A B Γ 1 (A) Γ 1 (B). Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 22 / 25

45 Matrix inequality Application Theorem 12 The principal inverse Γ 1 (x) of Γ(x) is operator monotone on [Γ(α), ); i.e., and hence for bounded self-adjoint operators A, B whose spectra are in [Γ(α), ) A B Γ 1 (A) Γ 1 (B). Proof A B implies that (A ti ) 1 (B ti ) 1 for t < Γ(α). From we have Γ 1 (A) Γ 1 (B). Γ 1 (x) = a + bx + Γ(α) ( 1 x t t t 2 +1 )dµ(t) Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 22 / 25

46 Application Then K 2 (x, y) := K log Γ(x) (x, y) = e K2(x,y) = { log Γ(x) log Γ(y) x y (x y) ( Γ(y) Γ(x) ) 1 e Γ (x) is infinitely divisible. Since Γ(x + 1) = xγ(x), Γ (x) Γ(x) x y (x y) Γ(x) (x = y) (x = y). Γ (1) Γ(1) = γ, Γ (m + 1) Γ(m + 1) = Γ (m) Γ(m) + 1 m, Γ(n) (n 1)! = Γ(m) (m 1)!. Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 23 / 25

47 matrix Application The following (n + 1) (n + 1) matrix is therefore not only p.s.d. but also infinitely divisible. ( e K 2 (i,j) ) = = e γ ( 1! 1! ) 1 (2!) 1 2 (3!) 1 3 (n!) 1 n ( 1! 1! ) 1 e γ 1 ( 2! 1! ) 1 ( 3! 1! ) 1 2 ( n! 1! ) n 1 (2!) 1 2 ( 2! 1! ) 1 e γ ( 3! 2! ) 1 ( n! 2! ) 1 n 2 (3!) 1 3 ( 3! 1! ) 1 2 ( 3! 2! ) 1 e γ ( n! (n!) 1 n ( n! 1 1! ) n 1 ( n! 1 2! ) n 2 ( n! 1 3! ) n 3 e γ n 1 3! ) n 3 Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 24 / 25

48 Application R. Bhatia, Infinitely divisible matrices, Amer. Math. Monthly, 113, (2006). R. Bahtia and H. Kosaki, Mean matrices and infinite divisibility, Linear Algebra Appl., 424, 36 54(2007). C. Berg, H. L. Pedersen, Pick functions related to the gamma function, Rocky Mountain J. of Math. 32(2002) A, Koranyi, On a theorem of Löwner and its connections with resolvents of selfadjoint transformations, Acta Sci. Math. 17, 63 70(1956). K. Löwner, Über monotone Matrixfunctionen, Math. Z. 38(1934) W. F. Donoghue, Monotone Matrix Functions and Analytic Continuation, Springer-Verlag, C. H. Fitzgerald, On analytic continuation to a Schlicht function, Proc. Amer. Math. Soc., 18, (1967). R. A. Horn, Schlicht mapping and infinitely divisible kernels, Pacific J. of Math. 38, (1971). R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, M. Uchiyama, Operator monotone functions, positive definite kernels and majorization, Proc. Amer. Math. 138(2010) M. Uchiyama, The principal inverse of the gamma function, PAMS (2012) Mitsuru Uchiyama (Shimane University) The principal inverse of the gamma function 2013/7 25 / 25